**Physics Notes - Herong's Tutorial Notes** - v3.24, by Herong Yang

Hamilton's Principle in Generalized Coordinates

This section provides a quick introduction to the Hamilton's Principle in Generalized Coordinates.

The nice thing about the Hamilton's Principle is that it it independent of coordinate systems. So the Hamilton's Principle stay the same: the true position function q(t) of a system is a stationary point of the Action S[q(t)] functional.

Mathematically, a stationary point of Action S[q(t)] functional is a function q(t), where the functional differential of S[q(t)] is zero. This can be expressed as:

```
d(S[q(t)]) / d(q(t)) = 0
# functional differential of S[q(t)]
or:
d(∫ L(q,q',t)dt) / dq = 0
# Replaced S with its definition
```

Hamilton's Principle is also called Stationary Action.

Table of Contents

Introduction of Frame of Reference

Introduction of Special Relativity

Time Dilation in Special Relativity

Length Contraction in Special Relativity

The Relativity of Simultaneity

Minkowski Spacetime and Diagrams

►Introduction of Generalized Coordinates

Generalized Coordinates and Generalized Velocity

Simple Pendulum Motion in Generalized Coordinates

►Hamilton's Principle in Generalized Coordinates

Lagrange Equations in Generalized Coordinates

Lagrange Equations on Simple Pendulum

What Is Legendre Transformation

Hamilton Equations in Generalized Coordinates